We turn now to the GPPE (REF ) coupled with the crust-rotation
equation (REF ). In Figs. REF  (a)-(d), we plot the
crust-potential isosurface in blue, with \(V_{\theta } = 450\) , and ten-level
isosurfaces of \((\nabla \times (\rho v))^2\)  for a representative set of
parameters and the times (a) \(t=0.06\) , (b) \(t=6.48\) , (c) \(t=7.38\) , and (d)
\(t=9.72\) . On average, the crust gains angular momentum from the
superfluid, hence, at long times, these vortices move outwards after losing
enough angular momentum to the crust (compare
Figs. REF  (a) and (d)). However, the time series of the
condensate and crust angular momenta, \(J_z\)  and \(J_c\) , respectively, are
complicated, and, as we show below, they display the signatures of
SOC [1]}, [2]}, [3]}, [4]}, [5]}, [6]}, [7]}, [8]}. We
illustrate this temporal evolution in Fig.REF ,
for the representative parameter values \(V_0 = 180\) , \(n_{\rm crust}=12\) , \(I_c= 0.01\) , \(r_{\rm crust} = 1.0\) , \(\Delta r_{\rm crust} = 0.15\) , \(\Omega _0 = 14\) , and
\(\alpha = 0.007\) . In Fig. REF (a) we plot, versus the scaled
time \(t\Omega _0\) , \((J_c/J_{c_0} - 14)\)  [blue curve], \(J_z/J_{c_0}\)  [red curve], and
\((J_c + J_z)/J_{c_0}\)  [green curve], where \(J_{c_0}\)  is the initial angular
momentum of the crust. This figure shows that, if we neglect the overall,
gentle decay of the total angular momentum We have noted above that, because we
use \(2 \pi \) -periodic coordinates to define \(J_z\) , the conservation of angular
momentum is only approximate, even if \(\alpha =0\) ; also, in neutron
stars, the crust angular momentum is only a few percent of the total angular
momentum [9]}., fluctuations of \(J_c\)  compensate for those in
\(J_z\) . In Figs. REF (b) and (c) we show expanded plots of
\(J_c/J_{c_0}\)  for \(0 < t\Omega _0 \lesssim 170\)  and \(80 \le t\Omega _0 \le 100\) ,
respectively, to illustrate the irregular nature of the time series of the
angular momentum of the crust.
<FIGURE>